Embedded newform 392.2.a.h.1.1

• Label: 392.2.a.h.1.1
• Level: $392$
• Weight: $2$
• Character: 392.1
• Self dual: yes
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{8})^+\)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			\(-1.41421\) of defining polynomial
Character	\(\chi\)	\(=\)	392.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} -2.82843 q^{5} +5.00000 q^{9} -4.00000 q^{11} +2.82843 q^{13} +8.00000 q^{15} +5.65685 q^{17} +2.82843 q^{19} +3.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} +5.65685 q^{31} +11.3137 q^{33} +10.0000 q^{37} -8.00000 q^{39} -5.65685 q^{41} -4.00000 q^{43} -14.1421 q^{45} -5.65685 q^{47} -16.0000 q^{51} +6.00000 q^{53} +11.3137 q^{55} -8.00000 q^{57} +2.82843 q^{59} -14.1421 q^{61} -8.00000 q^{65} +12.0000 q^{67} -8.48528 q^{75} +8.00000 q^{79} +1.00000 q^{81} +14.1421 q^{83} -16.0000 q^{85} -5.65685 q^{87} -16.0000 q^{93} -8.00000 q^{95} -5.65685 q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 10 * q^9 - 8 * q^11 + 16 * q^15 + 6 * q^25 + 4 * q^29 + 20 * q^37 - 16 * q^39 - 8 * q^43 - 32 * q^51 + 12 * q^53 - 16 * q^57 - 16 * q^65 + 24 * q^67 + 16 * q^79 + 2 * q^81 - 32 * q^85 - 32 * q^93 - 16 * q^95 - 40 * q^99

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−2.82843	−1.63299	−0.816497	−	0.577350i	\(-0.804087\pi\)
−0.816497	+	0.577350i				\(0.804087\pi\)
\(5\)	−2.82843	−1.26491	−0.632456	−	0.774597i	\(-0.717953\pi\)
			−0.632456	+	0.774597i	\(0.717953\pi\)
\(7\)	0	0
			\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
−0.603023	+	0.797724i				\(0.706037\pi\)
\(13\)	2.82843	0.784465	0.392232	−	0.919866i	\(-0.371703\pi\)
			0.392232	+	0.919866i	\(0.371703\pi\)
\(17\)	5.65685	1.37199	0.685994	−	0.727607i	\(-0.259367\pi\)
			0.685994	+	0.727607i	\(0.259367\pi\)
\(19\)	2.82843	0.648886	0.324443	−	0.945905i	\(-0.394823\pi\)
			0.324443	+	0.945905i	\(0.394823\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	5.65685	1.01600	0.508001	−	0.861357i	\(-0.330385\pi\)
			0.508001	+	0.861357i	\(0.330385\pi\)
\(37\)	10.0000	1.64399	0.821995	−	0.569495i	\(-0.192861\pi\)
			0.821995	+	0.569495i	\(0.192861\pi\)
\(41\)	−5.65685	−0.883452	−0.441726	−	0.897150i	\(-0.645634\pi\)
			−0.441726	+	0.897150i	\(0.645634\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	−5.65685	−0.825137	−0.412568	−	0.910927i	\(-0.635368\pi\)
			−0.412568	+	0.910927i	\(0.635368\pi\)
\(53\)	6.00000	0.824163	0.412082	−	0.911147i	\(-0.364802\pi\)
			0.412082	+	0.911147i	\(0.364802\pi\)
\(59\)	2.82843	0.368230	0.184115	−	0.982905i	\(-0.441058\pi\)
			0.184115	+	0.982905i	\(0.441058\pi\)
\(61\)	−14.1421	−1.81071	−0.905357	−	0.424650i	\(-0.860397\pi\)
			−0.905357	+	0.424650i	\(0.860397\pi\)
\(67\)	12.0000	1.46603	0.733017	−	0.680211i	\(-0.238112\pi\)
			0.733017	+	0.680211i	\(0.238112\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(79\)	8.00000	0.900070	0.450035	−	0.893011i	\(-0.351411\pi\)
			0.450035	+	0.893011i	\(0.351411\pi\)
\(83\)	14.1421	1.55230	0.776151	−	0.630548i	\(-0.217170\pi\)
			0.776151	+	0.630548i	\(0.217170\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	−5.65685	−0.574367	−0.287183	−	0.957876i	\(-0.592719\pi\)
			−0.287183	+	0.957876i	\(0.592719\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.a.h.1.1	✓	2
3.2	odd	2		3528.2.a.bj.1.2		2
4.3	odd	2		784.2.a.n.1.2		2
5.4	even	2		9800.2.a.bw.1.2		2
7.2	even	3		392.2.i.g.361.2		4
7.3	odd	6		392.2.i.g.177.1		4
7.4	even	3		392.2.i.g.177.2		4
7.5	odd	6		392.2.i.g.361.1		4
7.6	odd	2	inner	392.2.a.h.1.2	yes	2
8.3	odd	2		3136.2.a.bq.1.1		2
8.5	even	2		3136.2.a.bt.1.2		2
12.11	even	2		7056.2.a.cj.1.2		2
21.2	odd	6		3528.2.s.be.361.1		4
21.5	even	6		3528.2.s.be.361.2		4
21.11	odd	6		3528.2.s.be.3313.1		4
21.17	even	6		3528.2.s.be.3313.2		4
21.20	even	2		3528.2.a.bj.1.1		2
28.3	even	6		784.2.i.k.177.2		4
28.11	odd	6		784.2.i.k.177.1		4
28.19	even	6		784.2.i.k.753.2		4
28.23	odd	6		784.2.i.k.753.1		4
28.27	even	2		784.2.a.n.1.1		2
35.34	odd	2		9800.2.a.bw.1.1		2
56.13	odd	2		3136.2.a.bt.1.1		2
56.27	even	2		3136.2.a.bq.1.2		2
84.83	odd	2		7056.2.a.cj.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.a.h.1.1	✓	2	1.1	even	1	trivial
392.2.a.h.1.2	yes	2	7.6	odd	2	inner
392.2.i.g.177.1		4	7.3	odd	6
392.2.i.g.177.2		4	7.4	even	3
392.2.i.g.361.1		4	7.5	odd	6
392.2.i.g.361.2		4	7.2	even	3
784.2.a.n.1.1		2	28.27	even	2
784.2.a.n.1.2		2	4.3	odd	2
784.2.i.k.177.1		4	28.11	odd	6
784.2.i.k.177.2		4	28.3	even	6
784.2.i.k.753.1		4	28.23	odd	6
784.2.i.k.753.2		4	28.19	even	6
3136.2.a.bq.1.1		2	8.3	odd	2
3136.2.a.bq.1.2		2	56.27	even	2
3136.2.a.bt.1.1		2	56.13	odd	2
3136.2.a.bt.1.2		2	8.5	even	2
3528.2.a.bj.1.1		2	21.20	even	2
3528.2.a.bj.1.2		2	3.2	odd	2
3528.2.s.be.361.1		4	21.2	odd	6
3528.2.s.be.361.2		4	21.5	even	6
3528.2.s.be.3313.1		4	21.11	odd	6
3528.2.s.be.3313.2		4	21.17	even	6
7056.2.a.cj.1.1		2	84.83	odd	2
7056.2.a.cj.1.2		2	12.11	even	2
9800.2.a.bw.1.1		2	35.34	odd	2
9800.2.a.bw.1.2		2	5.4	even	2